Sum of oscillating sequences

Find two oscillating sequences such that the sum of those two sequences diverges to infinity or to minus infinity, if possible. I have been unable to find two such sequences. It is easy enough to find two oscillating sequences who sum converges, but not diverges. Any ideas? Thanks.

By "oscillating" I assume that you mean alternating, i.e., the sign changes like .

Think about what you want to do: a single alternating sequence can diverge, but it cannot have or as a limit, because of the sign changes. The idea would be to find a sequence whose subsequence of even terms approach but its odd subsequence (of negative terms) remain bounded. Then find a sequence whose odd terms approach but its even terms (here, negative as per requirement) remain bounded. The sum should do what you want.